Algebra Examples

$s = \sqrt{4 x^{2} + 6 y^{2}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x^{2} + 6 y^{2}}$ as ${(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}$ .

$s = {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (s) = \frac{d}{d x} ({(4 x^{2} + 6 y^{2})}^{\frac{1}{2}})$

Step 3

Since $s$ is constant with respect to $x$ , the derivative of $s$ with respect to $x$ is $0$ .

$0$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 4 x^{2} + 6 y^{2}$ .

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Step 4.1.1

To apply the Chain Rule, set $u_{1}$ as $4 x^{2} + 6 y^{2}$ .

$\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [4 x^{2} + 6 y^{2}]$

Step 4.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [4 x^{2} + 6 y^{2}]$

Step 4.1.3

Replace all occurrences of $u_{1}$ with $4 x^{2} + 6 y^{2}$ .

$\frac{1}{2} {(4 x^{2} + 6 y^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [4 x^{2} + 6 y^{2}]$

Step 4.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} {(4 x^{2} + 6 y^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [4 x^{2} + 6 y^{2}]$

Step 4.3

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{2} {(4 x^{2} + 6 y^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [4 x^{2} + 6 y^{2}]$

Step 4.4

Combine the numerators over the common denominator.

$\frac{1}{2} {(4 x^{2} + 6 y^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [4 x^{2} + 6 y^{2}]$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(4 x^{2} + 6 y^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [4 x^{2} + 6 y^{2}]$

Step 4.5.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(4 x^{2} + 6 y^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [4 x^{2} + 6 y^{2}]$

Step 4.6

Differentiate.

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Step 4.6.1

Move the negative in front of the fraction.

$\frac{1}{2} {(4 x^{2} + 6 y^{2})}^{- \frac{1}{2}} \frac{d}{d x} [4 x^{2} + 6 y^{2}]$

Step 4.6.2

Combine fractions.

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Step 4.6.2.1

Combine $\frac{1}{2}$ and ${(4 x^{2} + 6 y^{2})}^{- \frac{1}{2}}$ .

$\frac{{(4 x^{2} + 6 y^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [4 x^{2} + 6 y^{2}]$

Step 4.6.2.2

Move ${(4 x^{2} + 6 y^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}} \frac{d}{d x} [4 x^{2} + 6 y^{2}]$

Step 4.6.3

By the Sum Rule, the derivative of $4 x^{2} + 6 y^{2}$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [6 y^{2}]$ .

$\frac{1}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [6 y^{2}])$

Step 4.6.4

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}} (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 y^{2}])$

Step 4.6.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}} (4 (2 x) + \frac{d}{d x} [6 y^{2}])$

Step 4.6.6

Multiply $2$ by $4$ .

$\frac{1}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}} (8 x + \frac{d}{d x} [6 y^{2}])$

Step 4.6.7

Since $6$ is constant with respect to $x$ , the derivative of $6 y^{2}$ with respect to $x$ is $6 \frac{d}{d x} [y^{2}]$ .

$\frac{1}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}} (8 x + 6 \frac{d}{d x} [y^{2}])$

Step 4.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 4.7.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

$\frac{1}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}} (8 x + 6 (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y]))$

Step 4.7.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$\frac{1}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}} (8 x + 6 (2 u_{2} \frac{d}{d x} [y]))$

Step 4.7.3

Replace all occurrences of $u_{2}$ with $y$ .

$\frac{1}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}} (8 x + 6 (2 y \frac{d}{d x} [y]))$

Step 4.8

Multiply $2$ by $6$ .

$\frac{1}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}} (8 x + 12 (y \frac{d}{d x} [y]))$

Step 4.9

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$\frac{1}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}} (8 x + 12 y y')$

Step 4.10

Simplify.

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Step 4.10.1

Reorder the factors of $\frac{1}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}} (8 x + 12 y y')$ .

$(8 x + 12 y y') \frac{1}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}}$

Step 4.10.2

Multiply $8 x + 12 y y'$ by $\frac{1}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}}$ .

$\frac{8 x + 12 y y'}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}}$

Step 4.10.3

Factor $2$ out of $8 x$ .

$\frac{2 (4 x) + 12 y y'}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}}$

Step 4.10.4

Factor $2$ out of $12 y y'$ .

$\frac{2 (4 x) + 2 (6 y y')}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}}$

Step 4.10.5

Factor $2$ out of $2 (4 x) + 2 (6 y y')$ .

$\frac{2 (4 x + 6 y y')}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}}$

Step 4.10.6

Cancel the common factors.

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Step 4.10.6.1

Factor $2$ out of $2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}$ .

$\frac{2 (4 x + 6 y y')}{2 ({(4 x^{2} + 6 y^{2})}^{\frac{1}{2}})}$

Step 4.10.6.2

Cancel the common factor.

$\frac{2 (4 x + 6 y y')}{2 {(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}}$

Step 4.10.6.3

Rewrite the expression.

$\frac{4 x + 6 y y'}{{(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}}$

Step 4.10.7

Factor $2$ out of $4 x + 6 y y'$ .

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Step 4.10.7.1

Factor $2$ out of $4 x$ .

$\frac{2 (2 x) + 6 y y'}{{(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}}$

Step 4.10.7.2

Factor $2$ out of $6 y y'$ .

$\frac{2 (2 x) + 2 (3 y y')}{{(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}}$

Step 4.10.7.3

Factor $2$ out of $2 (2 x) + 2 (3 y y')$ .

$\frac{2 (2 x + 3 y y')}{{(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}}$

Step 5

Reform the equation by setting the left side equal to the right side.

$0 = \frac{2 (2 x + 3 y y')}{{(4 x^{2} + 6 y^{2})}^{\frac{1}{2}}}$

Step 6

Solve for $y'$ .

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Step 6.1

Set the numerator equal to zero.

$2 (2 x + 3 y y') = 0$

Step 6.2

Solve the equation for $y'$ .

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Step 6.2.1

Divide each term in $2 (2 x + 3 y y') = 0$ by $2$ and simplify.

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Step 6.2.1.1

Divide each term in $2 (2 x + 3 y y') = 0$ by $2$ .

$\frac{2 (2 x + 3 y y')}{2} = \frac{0}{2}$

Step 6.2.1.2

Simplify the left side.

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Step 6.2.1.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.2.1.1

Cancel the common factor.

$\frac{2 (2 x + 3 y y')}{2} = \frac{0}{2}$

Step 6.2.1.2.1.2

Divide $2 x + 3 y y'$ by $1$ .

$2 x + 3 y y' = \frac{0}{2}$

Step 6.2.1.3

Simplify the right side.

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Step 6.2.1.3.1

Divide $0$ by $2$ .

$2 x + 3 y y' = 0$

Step 6.2.2

Subtract $2 x$ from both sides of the equation.

$3 y y' = - 2 x$

Step 6.2.3

Divide each term in $3 y y' = - 2 x$ by $3 y$ and simplify.

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Step 6.2.3.1

Divide each term in $3 y y' = - 2 x$ by $3 y$ .

$\frac{3 y y'}{3 y} = \frac{- 2 x}{3 y}$

Step 6.2.3.2

Simplify the left side.

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Step 6.2.3.2.1

Cancel the common factor of $3$ .

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Step 6.2.3.2.1.1

Cancel the common factor.

$\frac{3 y y'}{3 y} = \frac{- 2 x}{3 y}$

Step 6.2.3.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{- 2 x}{3 y}$

Step 6.2.3.2.2

Cancel the common factor of $y$ .

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Step 6.2.3.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{- 2 x}{3 y}$

Step 6.2.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 2 x}{3 y}$

Step 6.2.3.3

Simplify the right side.

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Step 6.2.3.3.1

Move the negative in front of the fraction.

$y' = - \frac{2 x}{3 y}$

Step 7

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{2 x}{3 y}$